Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d \geq 2$

TL;DR

This work establishes nonperturbative bounds on two-to-two scattering of massive scalar particles in spacetime dimensions $d\ge 2$ by employing a primal S-matrix bootstrap that enforces full nonlinear unitarity for both fully nonperturbative and EFT amplitudes with cutoff $M$. It derives strong analytic and numerical positivity/dispersive bounds across $2\le d\le 4$, demonstrates sharper constraints than those from linearized unitarity or positivity alone, and systematically compares these bounds to perturbative $\phi^4$ theory, finding saturation at weak coupling. The authors develop a flexible numerical framework to bound observables such as $\lambda_{k,l}$ and $\Lambda_{k,l}$, extend the analysis to EFTs, and connect the results to pseudo-Goldstone EFTs and higher-dimensional theories, with publicly available data for reproducibility. Overall, the paper strengthens the S-matrix bootstrap program in higher dimensions, provides concrete EFT viability constraints, and highlights the deep link between perturbative amplitudes and nonperturbative bounds.

Abstract

We study two-to-two scattering amplitudes of a scalar particle of mass $m$. For simplicity, we assume the presence of $\mathbb{Z}_2$ symmetry and that the particle is $\mathbb{Z}_2$ odd. We consider two classes of amplitudes: the fully nonperturbative ones and effective field theory (EFT) ones with a cut-off scale $M$. Using the primal numerical method which allows us to impose full non-linear unitarity, we construct novel bounds on various observables in $2 \leq d \leq 4$ space-time dimensions for both classes of amplitudes. We show that our bounds are much stronger than the ones obtained by using linearized unitarity or positivity only. We discuss applications of our bounds to constraining EFTs. Finally, we compare our bounds to the amplitude in $φ^4$ theory computed perturbatively at weak coupling, and find that they saturate the bounds.

Figures (23)

• Figure 1: Analytic structure in the $s$ complex plane for a fixed value of $t$ of two classes of amplitudes considered in the literature: the left plot is for the nonperturbative amplitude and the right one is for the EFT amplitude. Here $m$ is the mass of the particle and $M$ is the EFT "cut-off". It is also assumed that $m\ll M$, thus we set $m=0$ in this case. No poles are present due to the assumed presence of $\mathbb{Z}_2$ symmetry.
• Figure 2: Contour integrals of the function $F(s')$ defined in equation \ref{['eq:function_F']}.
• Figure 3: Nonperturbative bound on the observables $(\tau_{0;0},\, \tau_{0;1})$ defined in \ref{['eq:description_3']} for various spacetime dimensions $d\in[2.4,4]$. Different colors represent different $d$. For each value of $d$, the allowed region is inside the corresponding "leaf" shape. The plot is built with $L_\text{max}=50$.
• Figure 4: Nonperturbative bound on the observables $(\tau_{0;0},\, \tau_{0;1})$ defined in \ref{['eq:description_3']} as a function of the spacetime dimension $d$. The plot is built with $L_\text{max}=50$.
• Figure 5: Dependence of left and right tips of the allowed regions in figure \ref{['fig:PA0_dPA0_Extrapolated']} on the spacetime dimension $d$. In the left plot, we have also indicated the $d=2$ result from equation \ref{['eq:boundLambda0']}, namely $(\tau_{0;0},\tau_{0;1})=(-8, 0)$. The dashed lines indicate how the $d>2$ results approach $d=2$ one. In the right plot, we see that the right tip in figure \ref{['fig:PA0_dPA0_Extrapolated']} approaches zero as we lower the spacetime dimension $d$.